# Ratio + Quantity

I am trying to create a metric which would tell us how good or bad a client is. Right now I am using a simple ratio: $$\frac{\text{number of bills paid on time}}{\text{number of bills}}.$$ This give me a percentage and it essentially tell us if a client always pays it bill on time ($0$ = never, $1$ = always). For instance, a client who paid only one bill on time is as good as a client who paid 20 bills on time with this metric.

The problem with this metric is that it doesn’t capture the number of bills the consumer had to pay. Does anybody has an idea on how I could capture these two information (paid on time + number of bills) into one single number?

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It also doesn't say how far behind they got; it treat a payment that's two days late as if it's just as bad as a payment that's two years late. – Michael Hardy Sep 9 '11 at 1:21
It also doesn't distinguish between clients based on how big the bills are. – Gerry Myerson Sep 9 '11 at 1:23
I suggest you take into account the value of the bills as in (total paid on time) / (total owned). – lhf Sep 9 '11 at 1:24

Before anyone can help you with this, there are some questions you have to decide. Do you want a higher number for a client who is 1 out of 1, or a client who is 19 out of 20? If 19/20 is better than 1/1, how about 18/20? 17/20? how bad does it have to be to be worse than 1/1?

Here's one possible answer: $${1+{\rm\ paid\ on\ time}\over2+{\rm\ number\ of\ bills}}$$ It has the following properties:

1. It's always (strictly) between 0 and 1.

2. If ${\rm paid\ on\ time}=0$ then it approaches zero as number of bills increases.

3. If all bills are paid on time then it approaches 1 as number of bills increases.

One bill, paid on time, gets evaluated as $.666\dots$, same as 13 bills out of 19 paid on time. If 13 out of 19 isn't as good as 1 out of 1, then you'll have to make some adjustment to the formula, replace the 1 and the 2 I put in there with some other numbers. In fact, I encourage you to experiment a bit, try different numbers in place of my 1 and 2, and see whether you can get a formula that matches your feelings about various situations.

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19 out of 20 is better than 1 out of 1. – Martin Sep 9 '11 at 1:53
OK, is 18/20 better than 1/1? – Gerry Myerson Sep 9 '11 at 1:54
Well, if a client has 18 bills paid on time I would suppose that it is still better than a single bill paid on time. – Martin Sep 9 '11 at 2:12
We're making progress, slowly. But you really have to work out where the break-even points are in order to get a formula that meets your needs. You have to know whether 17/20 is better than 1/1, whether 35/40 is better than 2/2, whether 20/30 is better than 7/10, etc., etc. I'll edit a suggestion into my answer, but only you can tell if it's really what you want. – Gerry Myerson Sep 9 '11 at 3:28
Thank you Gerry. This is exactly what I needed. – Martin Sep 9 '11 at 3:59